
the open substack \(\overline{\mathcal {J}}_{g,n}^{\operatorname{spl}}\)  parametrizes objects \((\mathcal {X}\rightarrow S, \mathcal {I})\)  as above such that the geometric fibers of \(\mathcal {I}\)  are simple sheaves;

for any \(\phi \in V_{g,n}\) , the open substack \(\overline{\mathcal {J}}_{g,n}(\phi )\)  corresponds to the stack defined in [1]} (using [2]}).

if \(\phi \in V_{g,n}\)  is general, then \(\overline{\mathcal {J}}_{g,n}(\phi )\)  is isomorphic to the Esteves' universal compactification \(\overline{\mathcal {J}}_{g,n}^{\mathcal {E}, ss}\)  (see [3]}) for a suitable choice of a universal vector bundle \(\mathcal {E}\)  on the universal family over \(\overline{\mathcal {M}}_{g,n}\)  (see [4]}).

the morphism \(\Phi :\overline{\mathcal {J}}_{g,n}\rightarrow \overline{\mathcal {M}}_{g,n}\)  is the morphism that sends objects \((\mathcal {X}\rightarrow S, \mathcal {I})\)  as above into \((\mathcal {X}\rightarrow S)\in \overline{\mathcal {M}}_{g,n}\) .


